# Course:Study Guide - Exam 1 - Bart-Fall08

## Part I: Essay type questions

The exam is 50 minutes long.
You should be thorough in your answer. Less than a half a page is likely not detailed enough of an explanation. I will ask some or all of the following questions:

• Compare and contrast the rozette, the frieze and the wallpaper symmetry groups

What are we looking for when we determine these symmetry groups? What types of symmetries do we look for in a rozette? In a frieze pattern? In a wallpaper pattern? I.e which patterns have reflections? Which ones have rotations? Of what degree? Which ones can have translations? In what directions? Anything else worth mentioning? Provide a (simple) example of each to show the difference. (Be thorough!)

• Show carefully that all parallelograms tessellate.

Be thorough in your explanations! How do you argue you can cover the plane without gaps and overlaps? What role do the angle measures play?

• Show carefully that all triangles tessellate.

Be thorough in your explanations! What role do the angle measures play? Be detailed in your explanation.

• Why are there exactly 3 regular tessellations?

First explain what a regular tessellation is. Then show carefully that there can only be three. Use the outline of the explorations we used. Give as much detail as possible.

• How does one in general use geometry to create Escher-like tessellations?

What kind of underlying geometric tessellations do we use? Does it matter which geometric tessellation we start with? For instance: if we want to create a tessellation with rotational symmetry, are we somewhat forced to start with a given geometric tessellation? If so, which one(s)? How do we create the tessellations by recognizable figures?

## Part II: Problems based on explorations and homework

• I will take some questions from the homework.
• I will take some questions from the explorations. See above for a list of all the explorations we have covered in class.